While TSA does have the authority to set size limits and did so in the post-9/11 era as a means of cutting down the volume it had to screen, that went away quickly and TSA has not since then gotten into the size limit business.
Carriers could certainly screen for size before the checkpoint and it's doable if a checkpoint only serves one carrier. UA, for instance, does this at LAX.
The bigger issue is that the cheapskate didn't beat the system, his carrier gave him a freebie. That's on the carrier. There is no prohibition on charging for gate-checked bags (UA does and AA will for their cheapest tickets).